Anna University Coimbatore 2009 B.E Two k with Answers for Transforms and Partial Differential Equations( Unit-2) - Question Paper

TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

 

ANNA UNIVERSITY (2 Marks) QUESTIONS WITH ANSWERS

 

UNIT-II

FOURIER TRANSFORM

TWO MARKS

 

1)        Write the Fourier transform pair.

Ans :Let f(x) be a given function defined in (-∞,+∞) then the fourier transform is given by corresponding inverse formula is given by . Equation (1) & (2) jointly called as fourier transform pair.

2)        If is the Fourier cosine transform of f(x), prove that the Fourier cosine transform of f(ax) is .

Ans : we know that .

Put then

3)        If F(S) is Fourier transform of f(x), write the Fourier transform of f(x)cos(ax) in terms of F.

Ans:

4)        State the Convolution theorem for Fourier transforms.

Ans: The Fourier transform of the convolution of is the product of their Fourier transform. .

 

 

 

5)        If F(S) is Fourier transform of f(x), then find the Fourier transform of f(x-a).

Ans :

 

6)        If is the Fourier Sine transform of f(x), show that .

Ans:

7)        Solve the integral equation . (April 2004)

Ans :

8)        Find the Fourier transform of f(x) if . (April 2004)

Ans :

9)        Find the Fourier cosine transform of . (Nov 2004)

Ans :

10)     Find a) and b) in terms of the Fourier transform of f

(b)

 

11)     State Fourier integral theorem. (April 2005) (May/June 2009)

Ans :

12)     Find the Fourier Sine transform of . (April 2005),(Dec 2008)

Ans :

13)     Find the Fourier transform of . (Nov/Dec 2005)

 

Ans:

 

 

 

14)     Find the Fourier cosine integral representation of .

Ans :

15)     Find the Fourier Sine transform of f(x)= . (May 2006)

Ans :

16)     Prove that , a>0. (May 2006)

Ans :

17)     If then give the value of . (May 2006)

Ans : Sama as 16

18)     Find the Fourier transform of . (May 2006)Ans :

 

 

 

19)     Find the Fourier cosine transform of f(x) defined as

. (Nov/Dec 2006) Ans :

20)     P.T where . (M/J 2007)

Ans :

21)     Write down the Fourier cosine transform pair formulae. (M/J 2007)

Ans:

 

 

22)     If prove that . (N/D 2003),(Nov 2005)

Ans :

23)     Prove that if , then (Dec 2008)

Ans :

 

 

24)     Find the Fourier transform of defined by

.(Dec 2008)

Ans:

 

 

 

 

25)     Find the Fourier Cosine transform of . (Dec 2008)

Ans :

26)     If , prove that . (Dec 2008)

Ans : By property,

27)     Find the Fourier sine transform of . (Dec 2008)

Ans : We know that ,

28)     State Parsevals identity on complex Fourier transforms. (Dec 2008)

 

 

29)     If then prove that . (Dec 2008)

Ans :

30)     State Modulation theorem in Fourier transform. (Dec 2008)

Ans :

31)     Give a function which is self reciprocal under Fourier sine and cosine transforms. (Dec 2008)

Ans :

32)     If , then prove that . (Dec 2008)

Ans :

 

 

 

 

****************************************